# Central Limit Theorem

**Type:** Theorem  
**Domain:** Probability & Statistics  
**Codex URL:** /mathematics/probability-statistics/central-limit-theorem/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Sum/average of many independent random variables tends toward normal distribution regardless of original distribution.

## Formal statement (Lindeberg-Lévy)
Zₙ = (ΣXᵢ − nμ)/(σ√n) → N(0,1) as n→∞

## Historical development
de Moivre (1733) — binomial special case. Term "central limit theorem" coined by Pólya (1920).

## Generalisations
Lyapunov (1901) and Lindeberg (1922) — relaxed to independent-but-not-identical distributions.

## Berry-Esseen theorem
Quantifies rate of convergence for finite n (Berry 1941, Esseen 1942).

## Failure case: Cauchy distribution
No finite mean/variance — sums remain Cauchy-distributed, never converge to normal. Shows CLT conditions are genuinely necessary.

## Sources
### Tier 1
- Feller, W. (1968/1971). *An Introduction to Probability Theory*.
- Billingsley, P. (1995). *Probability and Measure*. 3rd ed.
### Tier 2
- Stigler, S.M. (1986). *The History of Statistics*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*